Numerical renormalization group calculations of the magnetization of Kondo impurities with and without uniaxial anisotropy

We study a Kondo impurity model with additional uniaxial anisotropy $D$ in a nonzero magnetic field $B$ using the numerical renormalization group (NRG). The ratio $g_e/g_S$ of electron and impurity $g$ factor is regarded as a free parameter and, in particular, the special cases of a “local” ($g_e=0$) and “bulk” ($g_e=g_S$) field are considered. For a bulk field, the relationship between the impurity magnetization $\mathcal{M}$ and the impurity contribution to the magnetization $M_{\text{imp}}$ is investigated and it is shown that $\mathcal{M}$ and $M_{\text{imp}}$ are proportional to each other for fixed coupling strength. Furthermore, we find that the $g$-factor ratio effectively rescales the magnetic field argument of the zero-temperature impurity magnetization. In case of an impurity with $D=0$ and $g_e=g_S$, it is demonstrated that at zero temperature $\mathcal{M}\left(B\right)$, unlike $M_{\text{imp}}\left(B\right)$, does not display universal behavior. With additional “easy-axis” anisotropy, the impurity magnetization is “stabilized” at a $D$-dependent value for $k_{B}T\ll g_S{\mu }_{B}B\ll \left|D\right|$ and, for nonzero temperature, is well described by a shifted and rescaled Brillouin function on energy scales that are small compared to $\left|D\right|$. In the case of “hard-axis” anisotropy, the magnetization curves can feature steps which are due to field-induced pseudo-spin-$\frac{1}{2}$ Kondo effects. For large hard-axis anisotropy and a local field, these screening effects are described by an exchange-anisotropic spin-$\frac{1}{2}$ Kondo model with an additional scattering term that is spin dependent (in contrast to ordinary potential scattering). In accordance with the observed step widths, this effective model predicts a decrease of the Kondo temperature with every further step that occurs upon increasing the field. Our study is motivated by the question as to how the magnetic properties of a deposited magnetic molecule are modified by the interaction with a nonmagnetic metallic surface.

Figure 1

Main plots: Impurity contribution to the magnetization $M_{\text{imp}}$ and impurity magnetization $\mathcal{M}$ as a function of magnetic field for $g_e=g_S$, three different couplings $\rho J$, and for impurity spin (a) $S=\frac{1}{2}$, (b) $S=1$, and (c) $S=\frac{3}{2}$. The temperature is $k_{B}T/W\approx 1.54\times {10}^{-15}\approx 0$ and the field is rescaled using $k_B{T}_H$. In case of $S=1$, part of the universal BA solution for $M_{\text{imp}}(x,T=0)$ is missing in the regime $g_S{\mu }_{B}B\lesssim k_B{T}_H$ (Ref. 73). Upper left insets show NRG results for $M_{\text{imp}}\left(B\right)$ at $T\approx 0$ and finite temperature as a function of magnetic field, now expressed in units of $W$. $M_{\text{imp}}\left(B\right)$ for $J=0$ is also computed using NRG and resembles the magnetization of the free spin (Ref. 74). Thermal energies increase from left to right and range from $1.79\times {10}^{-6}W$ [(a)] or $1.95\times {10}^{-12}W$ [(b) and (c)] to $6.79\times {10}^{-3}W$. Results for adjacent temperatures are calculated using truncated Wilson chains whose lengths differ by five lattice sites. Lower right insets show a closeup of the magnetization curves for (b) low fields and (c) high fields along with data points for $M_{\text{imp}}$ that are multiplied by a coupling-dependent constant $\gtrsim 1$.

Figure 2

Impurity magnetization $\mathcal{M}$ for different anisotropy parameters $D<0$ (easy-axis anisotropy) as a function of magnetic field for $k_{B}T/W\approx 1.54\times {10}^{-15}\approx 0$, coupling strength $\rho J=0.07$, and impurity spin (a) $S=1$, (b) $S=\frac{3}{2}$, and (c) $S=2$. We compare with the magnetization of an impurity with $D=0$ (solid line). As before, equal $g$ factors of electrons and impurity are assumed. Vertical lines mark the respective value of $k_B{T}_H/W$, which is determined by fitting the universal Bethe ansatz solution for $M_{\text{imp}}(T=0)$ in case of $D=0$. For $\rho J=0.07$ and $S=2$, we find $k_B{T}_H/W\approx 6.8\times {10}^{-7}$.

Figure 3

Impurity magnetization $\mathcal{M}$ as a function of magnetic field for anisotropy $D/W=-{10}^{-3}$ and nonzero temperature $k_{B}T/W\approx 1.03\times {10}^{-6}$ for $S=1$, $k_{B}T/W\approx 1.28\times {10}^{-8}$ for $S=\frac{3}{2}$, and $k_{B}T/W\approx 1.58\times {10}^{-10}$ for $S=2$. Note that $\mathcal{M}$ is not saturated for any field in the plot range (cf. Fig. 2). Open symbols represent fits using a rescaled and shifted Brillouin function ${\tilde{B}}_S\left(x\right)=\gamma B_S\left(\eta x\right)$, and solid (green) lines fits using a rescaled and shifted Langevin function $\tilde{L}\left(x\right)=\gamma L\left(\eta x\right)$.

Figure 4

Energy levels with magnetic quantum numbers $M$ (upper panels) and magnetization (lower panels) as a function of magnetic field for an anisotropic spin, described by Hamiltonian (6) with $D>0$ (hard-axis anisotropy), with (a) $S=1$, (b) $S=\frac{3}{2}$, and (c) $S=2$. For the magnetization curves and fields larger than the respective saturation field, temperature increases from top to bottom.

Figure 5

Impurity magnetization $\mathcal{M}$ for varying hard-axis anisotropy $D>0$ as a function of magnetic field for temperature $k_{B}T/W\approx 1.54\times {10}^{-15}\approx 0$, coupling strength $\rho J=0.07$, equal $g$ factors, and impurity spin (a) $S=1$, (b) $S=\frac{3}{2}$, and (c) $S=2$. The value of $D$ increases from bottom to top or from left to right, respectively. Note the rescaling of the Zeeman energy $g_S{\mu }_{B}B$ by $D$. Circles indicate those magnetization curves for which $\left|\right(D-k_B{T}_H)/W|$ is minimal.

Figure 6

Impurity magnetization $\mathcal{M}$ as a function of magnetic field for impurity spin $S=1$ and anisotropy parameter (a) $D/W={10}^{-2}$, (b) $D/W={10}^{-6}$, and (c) $D/W={10}^{-7}$ (solid lines, cf. Fig. 5). Dashed lines show $\mathcal{M}\left(B\right)$ for $D=0$ and $k_{B}T/W\approx 1.54\times {10}^{-15}\approx 0$. For the light gray lines, the approximate value of $k_{B}T/W$ increases from left to right from (a) $4.35\times {10}^{-4}$ to $6.10\times {10}^{-2}$, (b) $1.99\times {10}^{-7}$ to $6.79\times {10}^{-3}$, and (c) $4.73\times {10}^{-10}$ to $6.79\times {10}^{-3}$. Adjacent finite-temperature curves in plots (b) and (c) are calculated using truncated Wilson chains whose lengths differ by either four or five lattice sites. Thin vertical lines indicate the respective Zeeman energy $g_S{\mu }_{B}B=D$. For the chosen coupling strength $\rho J=0.07$ and $D=0$, we have $k_B{T}_H/W\approx 3.39\times {10}^{-7}$ according to Table .

Figure 7

Impurity magnetization $\mathcal{M}$ as a function of magnetic field for impurity spin $S=\frac{3}{2}$ and anisotropy parameter (a) $D/W={10}^{-2}$, (b) $D/W={10}^{-6}$, and (c) $D/W={10}^{-10}$ (solid lines, cf. Fig. 5). As before, dashed lines show $\mathcal{M}\left(B\right)$ for $D=0$ and $k_{B}T/W\approx 0$. For the light gray lines, the approximate value of $k_{B}T/W$ increases from left to right from (a) $1.61\times {10}^{-5}$ to $1.05\times {10}^{-1}$, (b) $3.45\times {10}^{-7}$ to $6.79\times {10}^{-3}$, and (c) $1.01\times {10}^{-11}$ to $6.79\times {10}^{-3}$. Adjacent finite-temperature curves in plots (b) and (c) are again calculated using truncated Wilson chains whose lengths differ by either four or five lattice sites. Magnetic fields satisfying $g_S{\mu }_{B}B=nD$ with $n=1,2$ are highlighted by thin vertical lines. According to Table , $k_B{T}_H/W\approx 4.55\times {10}^{-7}$ for $\rho J=0.07$ and $D=0$.

Figure 8

Impurity magnetization $\mathcal{M}$ as a function of magnetic field for impurity spin $S=2$ and anisotropy parameter (a) $D/W={10}^{-2}$, (b) $D/W={10}^{-6}$, and (c) $D/W={10}^{-8}$ (solid lines, cf. Fig. 5). Dashed lines again show $\mathcal{M}\left(B\right)$ for $D=0$ and $k_{B}T/W\approx 0$. For the light gray lines, the approximate value of $k_{B}T/W$ increases from left to right from (a) $7.54\times {10}^{-4}$ to $1.05\times {10}^{-1}$, (b) $3.45\times {10}^{-7}$ to $6.79\times {10}^{-3}$, and (c) $5.26\times {10}^{-11}$ to $6.79\times {10}^{-3}$. Adjacent finite-temperature curves in plots (b) and (c) are calculated using truncated Wilson chains whose lengths differ by four to six lattice sites. Thin vertical lines indicate magnetic fields satisfying $g_S{\mu }_{B}B=nD$, with $n=1,2,3$. For $\rho J=0.07$ and $D=0$, we obtain $k_B{T}_H/W\approx 6.8\times {10}^{-7}$.

Figure 9

Impurity contribution to (a) the magnetization $M_{\text{imp}}$, (b) the entropy $S_{\text{imp}}$, and (c) the effective moment $T{\chi }_{\text{imp}}$ as a function of magnetic field for impurity spin $S=3$, hard-axis anisotropy $D/W={10}^{-3}$, and several temperature values. In plot (a), the dashed curve shows $M_{\text{imp}}\left(B\right)$ for $k_{B}T/W\approx 0$, and for the curves in plots (b) and (c) temperature increases from bottom to top. Thin vertical lines indicate the (equal) peak positions for $S_{\text{imp}}\left(B\right)$ and $T{\chi }_{\text{imp}}\left(B\right)$. As before, the coupling strength is chosen as $\rho J=0.07$ and equal $g$ factors are used.

Figure 10

Main plots: Impurity contribution to (a) the entropy $S_{\text{imp}}$, (b) the effective moment $T{\chi }_{\text{imp}}$, and (c) the magnetic susceptibility ${\chi }_{\text{imp}}$ as a function of temperature for impurity spin $S=3$ and hard-axis anisotropy $D/W={10}^{-3}$. The chosen nonzero magnetic fields correspond to the peak positions for $S_{\text{imp}}\left(B\right)$ and $T{\chi }_{\text{imp}}\left(B\right)$ according to Fig. 9. Also shown are NRG results for vanishing coupling strength (dashed lines) and for $D=0$ (dashed-dotted lines). Solid vertical lines delimit the temperature range considered in Fig. 9, whereas dashed vertical lines indicate the thermal energy for which $k_{B}T=D$. In the insets, data from the main plots are presented for a reduced temperature range.

Figure 11

Kondo temperature at the ELC field (cf. Appendix ) for the effective Hamiltonian with $g_e=0$ as a function of the scattering parameter $\kappa$ (corresponding to a pair of onsite energies ${\epsilon }_{0\downarrow }=-{\epsilon }_{0\uparrow }>0$ for the zeroth site of the Wilson chain) for fixed $J_{\parallel }$ and three values of the coupling parameter $J_{\perp }>J_{\parallel }$. Numbers in parentheses denote the respective ELC field and lines are intended as a guide to the eye. Pessimistic error bars would be smaller than the symbol size. Additional crosses indicate the Kondo temperature for parameters which, according to Eqs. (24) and (25), correspond to the given values of the quantum numbers $S$ and $M$. Small dots mark $T_K$ for the case of potential scattering, i.e., for spin-independent onsite energies and zero magnetic field. The Kondo temperature given in the upper right corner refers to the exchange-isotropic case without scattering term.

Figure 12

Negative value of the ELC field for a set of parameter combinations of the effective Hamiltonian with $g_e=0$ which, according to Eqs. (24) to (26), correspond to the indicated spin quantum numbers $S$, magnetic quantum numbers $0\le M<S$, and coupling parameters $J$. Projections of the data points onto two planes are shown as small dots and lines connect points belonging to the same values of $S$ and $J$. Pessimistic error bars would be smaller than the symbol size.

Figure 13

Kondo temperature at the respective ELC field for the same set of parameter combinations of the effective Hamiltonian with $g_e=0$ as in Fig. 12. Again, pessimistic error bars would be smaller than the symbol size. Three small crosses in the plane spanned by $k_B{T}_K^{\text{ELC}}/W$ and $J/W$ indicate the value of the Kondo temperature for the exchange-isotropic case without scattering term [as inferred from Table using Eq. (13)].

Figure 14

Limiting value for $T\to 0$ of (a) the impurity contribution to the magnetization $M_{\text{imp}}$ and (b) the impurity magnetization $\mathcal{M}$ as a function of the coupling strength $\rho J$ for impurity spin $S=1$, $g_e=g_S$, and several magnetic field values. The vertical lines mark a coupling strength of $\rho J^{\prime }\approx 0.276$ for which we have checked by comparing with the respective Bethe ansatz solution shown in Fig. 1 that $M_{\text{imp}}(B,T\approx 0)$ still exhibits universal behavior (a fit of the BA curve gives $k_B{T}_H/W\approx 8.85\times {10}^{-2}$). Remaining lines are intended as a guide to the eye.

Figure 15

Limiting value for $T\to 0$ of (a) the impurity contribution to the magnetization and (b) the impurity magnetization as a function of the coupling strength $\rho J$ as in Fig. 14, but here for impurity spin $S=\frac{3}{2}$. $M_{\text{imp}}(B,T\approx 0)$ again shows universal behavior for $\rho J^{\prime }\approx 0.276$ and a fit of the corresponding universal BA curve gives $k_B{T}_H/W\approx 3.05\times {10}^{-1}$.

Figure 16

Impurity magnetization $\mathcal{M}$ as a function of magnetic field for impurity spin $S=1$, hard-axis anisotropy $D/W={10}^{-4}$, coupling strength $\rho J=0.07$, and several values of the $g$-factor ratio $g_e/g_S$ [cf. Fig. 5a for $g_e=g_S$]. For equal $g$ factors, the impurity contribution to the magnetization $M_{\text{imp}}\left(B\right)$ is shown, too. The temperature is $k_{B}T/W\approx 1.54\times {10}^{-15}\approx 0$ in all cases.

Figure 17

Proportionality factor $\alpha$ appearing in Eq. (15) as a function of the transverse coupling strength $\rho J_{\perp }$ for impurity spin $S=\frac{1}{2}$ and three values of the longitudinal coupling $\rho J_{\parallel }$. Dashed horizontal lines mark the values of $\alpha$, given by the numbers in parentheses, that are predicted by Eq. (C7) for $\rho J_{\perp }\ll 1$. Open symbols indicate the proportionality factors for the exchange-isotropic case (i.e., $J_{\parallel }=J_{\perp }$), which are also found in Table . As before, $\alpha$ has been averaged over magnetic fields $g_S{\mu }_{B}B/W\in [{10}^{-13},{10}^{-1}]$ for $k_{B}T/W\approx 1.54\times {10}^{-15}\approx 0$. The corresponding standard deviations would amount to error bars smaller than the symbol size.

Figure 18

Impurity magnetization (in units of $g_S{\mu }_B$) for the effective Hamiltonian with $g_e=0$ and parameters according to Eqs. (24) to (26) versus the relative magnetic field for $k_{B}T/W\approx {10}^{-15}\approx 0$. The results have been calculated using the discretization schemes by (a) Yoshida et al. [with correction factor (Refs. 17, 18, and 84) $A_{\Lambda }$] (Refs. 81, 82, 83), (b) Campo and Oliveira (Ref. 84), and (c) Žitko and Pruschke (Refs. 70 and 71). For each discretization scheme, results are presented for three values of the discretization parameter $\Lambda$ and 16 values of the twist parameter $z$ (i.e., $z_i=i/16$ with $i\in \{1,2,...,16\}$). In plot (c), data points are vertically offset to enhance legibility. Numbers in parentheses denote the ELC field ${\tilde{B}}_{\text{ELC}}$ and the Kondo temperature $T_K^{\text{ELC}}$ at the ELC field, respectively (cf. main text).